Rank 2 stable sheaves with odd determinant on Fano threefolds of genus 9

TL;DR

This paper investigates rank-2 stable sheaves with odd determinant on Fano threefolds of genus 9, revealing their moduli spaces' structure and connections to Brill-Noether loci and Picard varieties.

Contribution

It establishes a birational correspondence between moduli spaces of sheaves on the threefold and Brill-Noether loci on an associated surface, and describes the moduli space for c_2=7 as a blow-up of a Picard variety.

Findings

A component of the moduli space is birational to a Brill-Noether locus on the associated surface.

The moduli space M_X(2,1,7) is isomorphic to the blow-up of Pic^2(Gamma).

The study uses Kuznetsov's integral functor to analyze the sheaves.

Abstract

By the description due to Mukai and Iliev, a smooth prime Fano threefold X of genus 9 is associated to a surface P(V), ruled over a smooth plane quartic Gamma. We use Kuznetsov's integral functor to study rank-2 stable sheaves on X with odd determinant. For each c_2 \geq 7, we prove that a component of their moduli space M_X(2,1,c_2) is birational to a Brill-Noether locus of bundles on Gamma having enough sections when twisted by V. Moreover we prove that M_X(2,1,7) is isomorphic to the blowing-up of the Picard variety Pic^2(Gamma) along the curve parametrizing lines contained in X.